Search for Gamma-ray Emission from Accretion Flares of Tidal Disruption Events Possibly Associated with the IceCube Neutrinos

Abstract

Outflows or disk-coronas generated in tidal disruption events (TDEs) of supermassive black holes have been suggested as possible sites of high-energy neutrinos. Three TDEs (AT2019dsg, AT2019fdr and AT2019aalc) have been claimed to be associated with high-energy astrophysical neutrinos in multi-messenger follow-ups. No GeV photons have been detected accompanying the neutrino for the three sources. In this work, we searched for the high-energy gamma-ray emission from a larger sample of TDE candidates observed by the Zwicky Transient Facility (ZTF). No significant GeV emission was observed, and the upper limits of the gamma-ray emission flux are reported. We then performed a stacking analysis for the sample sources and found that the collective gamma-ray emission of this class of sources was also not bright enough to be detected by the Fermi Large Area Telescope (Fermi-LAT). The nondetection of the high-energy gamma-ray emission from the sample TDEs could be due to the fact that the high-energy gamma rays are absorbed by soft photons in the source. Using a model-based hypothesis, the upper limit on the emission radius of the neutrino production is obtained for these TDEs: $R<{10}^{16}$ cm for typical TDE parameter values.

1. Introduction

The IceCube telescope has detected TeV-PeV neutrinos for the first time [1,2], which opens a unique window to explore the high-energy universe. While it is known that many of the neutrinos are of astrophysical origin, as indicated by the isotropic distribution, the nature of the incontrovertible sources is unknown despite numerous cross-correlation analyses between the IceCube point-source data and multi-wavelength catalogues (e.g., [3,4,5,6]), as well as rapid follow-up observations utilizing the real-time alert framework [7]. Thus far, a few candidates are of special interest: PKS B1424-418, NGC 1068 and GB6 J1542+6129 [8,9,10]. Especially, TXS 0506+056, a very powerful high-energy BL Lac object at a redshift of $z=0.3365$, is identified following the IceCube-170922A neutrino event for the first time by the IceCube cooperation [8,11]. Although these candidates exhibit multi-wavelength emission in positional and temporal coincidence with high-energy neutrino events, the obtained significance level is not high enough to claim a firm identification due to the poor angular resolution of order 1–10 degrees at observed energies for IceCube neutrinos.

These associations and other conceptual arguments suggest that the neutrino flux may arise from a mixture of different astrophysical populations. The potential astrophysical sources that could produce high-energy neutrinos include star-forming galaxies (SFGs) [12,13,14,15,16,17], gamma-ray bursts (GRBs) [18,19,20], active galactic nuclei (AGN; blazars or cores) [21,22,23,24] and tidal disruption events (TDEs) [25,26,27,28]. In the various candidates mentioned above, TDEs have received more attention in the last several years. TDEs are rare transients when a star ventures too close to a black hole. Studies have suggested that TDEs are the possible sources of ultra-high-energy cosmic rays (UHECRs) [25,29] and high-energy neutrinos [25,26], which holds in particular for the subset of TDEs with successful or choked relativistic jets [26].

Recently, three TDE candidates (AT2019dsg, AT2019fdr and AT2019aalc) discovered by ZTF have been reported as the counterparts of high-energy astrophysical neutrinos in multi-messenger follow-up programs [30,31,32]1. Observationally, the three observed neutrino-emitting TDEs show common characteristics over a wide range of wavelengths, incorporating a bright black body spectrum in the optical–ultraviolet (OUV) band, infrared (IR) echo emission, as well as radio radiation from the synchrotron spectrum of non-thermal high-energy electrons. In order to understand these phenomena, many scenarios such as TDE jets [34,35], outflow-cloud interactions [36], the core region of AGNs [30,37] or accretion flares from massive black holes [32] have been proposed. In these models, $p\gamma$ interactions between relativistic protons accelerated in the jet, accretion flows, corona or disk-driven winds and the intense OUV/X-ray radiation of the TDE produce the neutrinos. Or, protons accelerated by the bow shocks in the outflow generate PeV neutrinos by the $pp$ interaction with clouds.

No gamma-ray emission is measured using the Fermi-LAT for the three TDEs, which may be explained as the absorption by the OUV and/or the X-ray radiation field of the TDEs via the $\gamma \gamma$ annihilation. It should be noted that high-energy gamma rays accompanying neutrinos would produce the electron–positron pairs and hence would trigger the electromagnetic cascades and deposit their energies into the GeV band via the inverse Compton (IC) radiation. The $\gamma \gamma$ annihilation opacity depends on the viewing angle because the outflow wind has drastically different density and velocity profiles at different inclination angles [38]. GeV photons could be detectable for the Fermi-LAT under some certain employed parameters of an off-axis jet model (see Figure 2 in [34]). Other physical parameters, such as the CR loading factor in the hidden wind model, have significant effects on the GeV emission (see Figure 5 in [37]). Gamma-ray emission is also expected from the unbound debris of TDEs interacting with the surrounding medium [39]. The high-energy photons produced in this case would not suffer from the strong absorption. What is more, from the perspective of isotropic energy, the derived total bolometric energy reaches to $\sim {10}^{52}$ erg for the three TDEs, comparable to that in the GRB prompt emission and blazar flares, which are two typical GeV emitters.

The GeV emission, combined with observation at other bands, could not only constrain the TDE model, like the viewing angle, magnetic field and CR loading factor, but also be a key signature for neutrino production. Therefore, we will further search for the GeV emission from a larger sample of TDEs with accretion flares, similar to the above three events. The sample is a new population of TDEs constructed based on the ZTF [40,41,42] and the Wide-field Infrared Survey Explorer (WISE) observation [43]; see van Velzen et al. [32] for more details. In addition, it is different from that in [44], which focuses on the high-energy emission from jetted TDEs. The sample here has not been observed in the direct signatures of the jets. Our work is organized as follows. We present the data analysis and results in Section 2 and Section 3, respectively. In Section 4, we discuss and summarize our findings.

2. Data Reduction

The set of sources selected for analysis is as given in extended table of van Velzen et al. [32]. The sample excludes ones with Galactic longitude $\left|b\right|<{20}^{\circ }$ to avoid contamination from diffuse gamma rays near the plane of the Milky Way. The final sample chosen for the Fermi-LAT data analysis therefore consisted of 52 sources.

2.1. Fermi-LAT Data Analysis

The LAT on board the Fermi mission is a pair-conversion instrument that is sensitive to GeV emission [45,46]. We collected Fermi-LAT data in the sky-survey mode from the start of the mission to 2022. Data were analyzed with the fermitools version 2.0.8. A binned maximum likelihood analysis was performed on a region of interest (ROI) with a radius ${10}^{\circ }$ centered on the “R.A.” and “decl.” of each source. Recommended event selections for data analysis were “FRONT+BACK” (evtype=3) and evclass=128. We applied a maximum zenith angle cut of $z_{\mathrm{zmax}}={90}^{\circ }$ to reduce the effect of the Earth albedo background. The standard gtmktime filter selection with an expression of (DATA_QUAL $>0\&\&$ LAT_CONFIG $==1$) was set. A source model was generated containing the position and spectral definition for all the point sources and diffuse emission from the 4FGL [47] within ${15}^{\circ }$ of the ROI center. The analysis included the standard galactic diffuse emission model ($\mathrm{gll}\_\mathrm{iem}\_\mathrm{v}07.\mathrm{fi}\mathrm{ts}$) and the isotropic component (iso_P8R3_SOURCE_V3_v1.txt), respectively. The former includes gamma-ray emission produced via interactions between CRs and interstellar matter, IC scattering of interstellar soft photons off CR electrons. In addition, the latter contains extra-galactic diffuse gamma rays, unresolved extra-galactic sources and residual (mis-classified) CR emission. These two files are designed to be used for point-source analysis, as suggested by the Fermi-LAT team. We binned the data in counts maps with a scale of $0.1^{\circ }$ per pixel and used 30 logarithmically spaced bins in energy. The energy dispersion correction was made when event energies extending down to 100 MeV were taken into consideration.

Because the 4FGL-DR3 catalog covered a full 12 years of data and we used a shorter time period (see the next section for details), the normalizations of the diffuse components and the spectral normalizations of catalogued LAT sources located within ${10}^{\circ }$ of the center of the ROI were kept free during the maximum likelihood fitting. In a few cases, we fixed or deleted some weak sources to obtain a convergent fit. If a new gamma-ray transient signal from a new point source emerged by building test statistic (TS) maps, we carried out the fit again, incorporating the new point source with a power-law spectrum $\frac{dN}{dE}=N_0{\left(\frac{E}{E_0}\right)}^{-\Gamma }$.

2.2. WISE Data Analysis

The WISE [43] telescope has been operating a repetitive all-sky survey since 2010, except for a gap between 2011 and 2013. The WISE telescope visits each location every half a year and takes >10 exposures during ∼1 day. Although initially four filters were used, most of the time, only two filters, named W1 and W2, were used. The central wavelengths of the two filters are $3.4$ and $4.6\mathsf{\mu }$m. Some sources are relatively faint and only marginally detected in any single-exposure image. Thus, we start from time-resolved coadds that were generated by Meisner et al. [48] by stacking the single-exposure images taken during each WISE visit. We performed point-spread function (PSF) fitting photometry on each coadd following Lang et al. [49], and during the fitting, the position was fixed to that from the optical survey. In this way, we obtained lightcurves sampled once every half a year at band of $3.4$ and $4.6\mathsf{\mu }$m for the sources in our sample. Based on the WISE lightcurves, we can obtain a reference time for Fermi-LAT data analysis, as discussed in the next section.

3. Results

3.1. Single-Source Analysis

Since the neutrinos arrive at the time after the optical peak observed by ZTF and before the IR peak observed by WISE, as shown in Figure 3 of van Velzen et al. [32], we defined three time intervals for the Fermi-LAT analysis. We selected 30 days before and after the WISE peak (${\Delta }_{\mathrm{WISE}}$), 30 days before and after the ZTF peak (${\Delta }_{\mathrm{ZTF}}$) and the duration between the ZTF peak and the WISE peak ($\Delta T_{\mathrm{ZW}}$) for the individual source in our sample. The distribution of $\Delta T_{\mathrm{ZW}}$ was from ∼60 to ∼650 days. As one can see in

Table 1. Upper limits at 95% confidence level on gamma-ray flux of each source for different time intervals.

Name	${\mathit{T}}_{\mathbf{peak}}$ 1	$\frac{\mathbf{\Delta }{\mathit{F}}_{\mathbf{IR}}}{{\mathit{F}}_{\mathbf{rms}}}$ 2	z	${\mathit{F}}_{\mathit{\gamma },\mathbf{\Delta }{\mathit{T}}_{\mathbf{WiSE}}}$	${\mathit{F}}_{\mathit{\gamma },\mathbf{\Delta }{\mathit{T}}_{\mathbf{ZTF}}}$	${\mathit{F}}_{\mathit{\gamma },\mathbf{\Delta }{\mathit{T}}_{\mathbf{ZW}}}$
				(phs cm${}^2$ s${}^{-1}$)	(phs cm${}^2$ s${}^{-1}$)	(phs cm${}^2$ s${}^{-1}$)
AT2019fdr	58,672.5	39.2	0.2666	<$2.25\times {10}^{-9}$	<$2.51\times {10}^{-9}$	<$1.77\times {10}^{-9}$
AT2019aalc	58,658.2	15.7	0.0356	<$4.17\times {10}^{-9}$	<$8.99\times {10}^{-9}$	<$3.07\times {10}^{-9}$
AT2018dyk	58,261.4	23.8	0.0367	<$2.08\times {10}^{-9}$	<$5.84\times {10}^{-9}$	<$1.08\times {10}^{-9}$
AT2019aame	58,363.2	12.3	–	<$1.15\times {10}^{-8}$	<$3.11\times {10}^{-9}$	<$4.57\times {10}^{-9}$
AT2018lzs	58,378.2	3.3	–	<$2.66\times {10}^{-9}$	<$3.62\times {10}^{-9}$	<$1.09\times {10}^{-9}$
AT2021aetz	58,390.3	47.5	0.0879	<$4.57\times {10}^{-9}$	<$2.35\times {10}^{-9}$	<$1.13\times {10}^{-9}$
AT2018iql	58,449.4	30.1	–	<$1.66\times {10}^{-8}$	<$1.86\times {10}^{-8}$	<$1.19\times {10}^{-8}$
AT2018jut	58,449.6	5	–	<$3.45\times {10}^{-9}$	<$2.47\times {10}^{-9}$	<$1.37\times {10}^{-9}$
AT2021aeue	58,475.1	4.9	–	<$4.03\times {10}^{-9}$	<$1.15\times {10}^{-8}$	<$2.02\times {10}^{-9}$
AT2019aamf	58,506.4	6.6	–	<$3.85\times {10}^{-9}$	<$5.31\times {10}^{-9}$	<$4.69\times {10}^{-10}$
AT2018kox	58,510.2	5.6	0.096	<$2.41\times {10}^{-9}$	<$4.00\times {10}^{-9}$	<$3.66\times {10}^{-9}$
AT2018lhv	58,513.5	32.3	–	<$4.54\times {10}^{-9}$	<$4.56\times {10}^{-9}$	<$2.97\times {10}^{-9}$
AT2019avd	58,534.3	67.5	0.0296	<$7.67\times {10}^{-9}$	<$7.44\times {10}^{-9}$	<$6.44\times {10}^{-10}$
AT2016eix	58,539.4	6.9	–	<$2.41\times {10}^{-9}$	<$2.72\times {10}^{-9}$	<$1.24\times {10}^{-9}$
AT2019aamg	58,540.5	8.3	–	<$4.61\times {10}^{-9}$	<$2.33\times {10}^{-9}$	<$2.14\times {10}^{-9}$
AT2021aeuf	58,556.4	15.6	–	<$1.33\times {10}^{-9}$	<$3.56\times {10}^{-9}$	<$1.86\times {10}^{-9}$
AT2020aezy	58,558.4	4.8	–	<$4.03\times {10}^{-9}$	<$2.35\times {10}^{-9}$	<$1.11\times {10}^{-9}$
AT2019aamh	58,582.5	7.7	–	<$3.73\times {10}^{-9}$	<$1.30\times {10}^{-8}$	<$1.46\times {10}^{-9}$
AT2019dll	58,605.2	6.8	0.101	<$1.46\times {10}^{-8}$	<$4.09\times {10}^{-9}$	<$2.18\times {10}^{-9}$
AT2018lof	58,608.2	4.1	0.302	<$8.47\times {10}^{-9}$	<$2.67\times {10}^{-9}$	<$1.92\times {10}^{-9}$
AT2019dqv	58,628.2	40.4	0.0816	<$1.70\times {10}^{-9}$	<$1.84\times {10}^{-9}$	<$5.48\times {10}^{-10}$
AT2019cyq	58,637.2	31.8	0.262	<$4.85\times {10}^{-9}$	<$2.07\times {10}^{-9}$	<$1.34\times {10}^{-9}$
AT2021aeug	58,641.2	4.6	–	<$1.75\times {10}^{-9}$	<$4.52\times {10}^{-9}$	<$5.13\times {10}^{-10}$
AT2019ihv	58,646.5	8.7	0.1602	<$3.46\times {10}^{-9}$	<$3.37\times {10}^{-9}$	<$1.68\times {10}^{-9}$
AT2019dzh	58,651.2	6.4	0.314	<$7.65\times {10}^{-9}$	<$2.02\times {10}^{-9}$	<$1.23\times {10}^{-9}$
AT2019kqu	58,652.2	6.1	0.174	<$2.26\times {10}^{-9}$	<$3.79\times {10}^{-9}$	<$1.59\times {10}^{-9}$
AT2020aezz	58,677.3	5.8	–	<$3.81\times {10}^{-9}$	<$4.30\times {10}^{-9}$	<$1.71\times {10}^{-9}$
AT2020afaa	58,678.2	7	–	<$3.32\times {10}^{-9}$	<$2.41\times {10}^{-9}$	<$6.84\times {10}^{-10}$
AT2019idm	58,682.2	25.2	0.0544	<$1.93\times {10}^{-9}$	<$5.96\times {10}^{-9}$	<$9.02\times {10}^{-10}$
AT2019ihu	58,709.5	6.2	0.27	<$2.16\times {10}^{-9}$	<$2.13\times {10}^{-9}$	<$2.00\times {10}^{-9}$
AT2019nna	58,717.4	27	–	<$1.09\times {10}^{-8}$	<$8.63\times {10}^{-9}$	<$2.90\times {10}^{-9}$
AT2019nni	58,732.2	4.9	0.137	<$2.95\times {10}^{-9}$	<$5.06\times {10}^{-9}$	<$1.93\times {10}^{-9}$
AT2021aeuk	58,733.1	7.3	0.235	<$3.26\times {10}^{-9}$	<$3.49\times {10}^{-9}$	<$1.02\times {10}^{-9}$
AT2019hdy	58,749.5	4	0.442	<$7.67\times {10}^{-9}$	<$6.96\times {10}^{-9}$	<$2.38\times {10}^{-9}$
AT2019pev	58,750.1	7.4	0.097	<$9.84\times {10}^{-9}$	<$5.82\times {10}^{-9}$	<$1.55\times {10}^{-9}$
AT2013kp	58,753.1	44.4	0.01499	<$3.32\times {10}^{-9}$	<$2.63\times {10}^{-9}$	<$2.70\times {10}^{-9}$
AT2019brs	58,758.1	9.6	0.3736	<$2.37\times {10}^{-9}$	<$5.68\times {10}^{-9}$	<$6.59\times {10}^{-10}$
AT2020afac	58,758.3	10.8	–	<$3.72\times {10}^{-9}$	<$9.51\times {10}^{-9}$	<$3.57\times {10}^{-9}$
AT2019wrd	58,764.3	7.6	–	<$7.07\times {10}^{-9}$	<$6.42\times {10}^{-9}$	<$1.01\times {10}^{-9}$
AT2021aeuh	58,789.5	3.9	0.0834	<$6.14\times {10}^{-9}$	<$4.63\times {10}^{-9}$	<$1.75\times {10}^{-9}$
AT2019msq	58,791.2	6.4	–	<$2.13\times {10}^{-9}$	<$6.44\times {10}^{-9}$	<$1.09\times {10}^{-9}$
AT2019qpt	58,798.3	13.8	0.242	<$3.10\times {10}^{-9}$	<$5.71\times {10}^{-9}$	<$3.25\times {10}^{-9}$
AT2020afad	58,802.2	3.7	–	<$2.01\times {10}^{-9}$	<$5.42\times {10}^{-9}$	<$9.71\times {10}^{-10}$
AT2019mss	58,811.6	20.8	–	<$1.14\times {10}^{-8}$	<$3.25\times {10}^{-9}$	<$3.30\times {10}^{-9}$
AT2019thh	58,851.1	72.2	0.0506	<$3.21\times {10}^{-9}$	<$2.69\times {10}^{-9}$	<$3.87\times {10}^{-9}$
AT2021aeui	58,860.3	6.2	–	<$4.22\times {10}^{-9}$	<$2.92\times {10}^{-9}$	<$4.09\times {10}^{-9}$
AT2020mw	58,867.3	6.8	–	<$1.75\times {10}^{-9}$	<$3.70\times {10}^{-9}$	<$2.53\times {10}^{-9}$
AT2020iq	58,878.1	24.5	0.096	<$7.01\times {10}^{-9}$	<$4.19\times {10}^{-9}$	<$1.05\times {10}^{-9}$
AT2019xgg	58,891.2	4.4	–	<$6.98\times {10}^{-9}$	<$4.66\times {10}^{-9}$	<$2.23\times {10}^{-9}$
AT2020atq	58,903.2	20.8	–	<$8.37\times {10}^{-9}$	<$3.82\times {10}^{-9}$	<$2.35\times {10}^{-9}$
AT2021aeuj	58,974.2	18.1	0.695	<$1.29\times {10}^{-8}$	<$4.27\times {10}^{-9}$	<$2.66\times {10}^{-9}$
AT2020hle	58,978.3	21	0.103	<$3.98\times {10}^{-9}$	<$2.74\times {10}^{-9}$	<$6.44\times {10}^{-10}$

¹ The peak time of ZTF lightcurve. ² The mean IR flux increase in the first year after the optical transient over the root mean square (rms) variability prior to the optical peak. $\Delta {\mathit{T}}_{\mathrm{WiSE}}$ and $\Delta {\mathit{T}}_{\mathrm{ZTF}}$ are 30 days before and after the peak time of WISE lightcurve and ZTF lightcurve, respectively. $\Delta {\mathit{T}}_{\mathrm{ZW}}$ includes the time interval between the peak time of ZTF and the peak time of WISE lightcurve.

, no one showed a significant gamma-ray emission with $TS<9$ on the different time intervals, yielding upper limits at 95% confidence level on the gamma-ray flux from these 52 sources.

For the three TDEs, AT2019dsg, AT2019fdr and AT2019aalc, the Fermi-LAT data analysis is studied, already covering a similar time window relative to the optical discovery and the neutrino arrival time [32]. In this time interval, there was no clear gamma-ray excess for the three sources. Here, we re-analyzed the Fermi-LAT data in the energy range 0.1–100 GeV, extending the time intervals from the launch of Fermi to 2022, with the evenly spaced binning of six months. All the searches yielded negative results; see the 95% confidence level upper limits in Figure 1.

Besides the three neutrino-coincident TDEs, we also made a thorough investigation of the gamma-ray emission from AT2013kp, a TDE in our sample with the smallest redshift $z=0.01499$ (corresponding to a luminosity distance 65 Mpc using the concordance cosmological parameters: ${\Omega }_{m,0}=0.286$ and $H_0=69.6$ km/s/Mpc). We chose similar linear bins with equal time widths of six months. We found no evidence of gamma-ray emission from any period and determined the corresponding flux upper limits reported in Figure 2.

3.2. Stacking Analysis

We analyzed in detail the population of the undetected sources in our sample to search for the collective emission from these objects. To reach this goal, we performed a stacking analysis following the method in [50], assuming the same power-law spectrum ($\Gamma$ = 2, $E_0$ = 1 GeV) for all sources. The stacking analysis can give a better sensitivity of the analysis by merging the observations of multiple sources. We summed the data (i.e., the count cubes in the analysis) of 52 sources together. The likelihood was evaluated by

$$ln\mathcal{L}=\sum _i{N}_i\ln M_i-M_i-ln{N}_i!$$

(1)

where $M_i$ and $N_i$ are the model-expected counts and observed counts in each pixel, respectively, and the index i runs over all energy and spatial bins. The $M_i$ is obtained using gtmodel in Fermitool software. For our stacking analysis, $Mi={\sum }_k{m}_{i,k}$ and $N_i={\sum }_k{n}_{i,k}$ with index k summing over 52 sources. We obtained the upper limits at 95% confidence level when $ln\mathcal{L}$ changed by 1.35. The results are presented in

Table 2. Results of the stacking analysis for all the sample sources for different time intervals.

Interval	TS	${\mathit{F}}_{\mathit{\gamma }}$	${\mathit{F}}_{\mathit{\gamma }}$
(Days)		(phs cm${}^2$ s${}^{-1}$)	(erg cm${}^2$ s${}^{-1}$)
$\Delta T_{\mathrm{WISE}}$	0.23	<$3.61\times {10}^{-10}$	<$4.00\times {10}^{-13}$
$\Delta T_{\mathrm{ZTF}}$	0.00	<$1.93\times {10}^{-10}$	<$2.14\times {10}^{-13}$
$\Delta T_{\mathrm{ZW}}$	0.00	<$1.72\times {10}^{-10}$	<$1.91\times {10}^{-13}$

Notes: $\Delta {\mathit{T}}_{\mathrm{WiSE}}$, $\Delta {\mathit{T}}_{\mathrm{ZTF}}$ and $\Delta {\mathit{T}}_{\mathrm{ZW}}$ as defined in Table 1.

. The upper limits obtained from the stacking analysis are nearly one order of magnitude below the averaged upper limits of the individual source, implying that no sub-threshold gamma-ray excess appears around the position of each source. In other words, the stacking results are compatible with the prediction derived from the analysis of background fluctuations. Note that our sample is contaminated by a few AGNs, which could introduce a slight but insignificant difference in the results of the stacking analysis.

4. Discussion and Conclusions

The physical processes in TDEs are not fully understood, e.g., the radius where various processes take place is highly uncertain. X-ray photons are usually observed in TDEs. GeV gamma-ray emission produced directly in ${\pi }^0$ decay or through the related cascade process will be absorbed by soft photons, i.e., the annihilation opacity between a high-energy photon ($E_h$) and a low-energy photon ($E_t$) in the source ${\tau }_{\gamma \gamma }>1$. For a high-energy photon with $E_h=5$ GeV typically detected by the Fermi-LAT, the energy of low-energy target photons ($E_t$) is $E_t\ge 2{\left(m_e{c}^2\right)}^2/E_h=100{(E_h/5\mathrm{GeV})}^{-1}$ eV, which would just be provided by the observed soft X-ray thermal emission [32]. The optical depth ${\tau }_{\gamma \gamma }$ is given by

$${\tau }_{\gamma \gamma }={\sigma }_{\gamma \gamma }nR$$

(2)

where ${\sigma }_{\gamma \gamma }\approx 1/4{\sigma }_{\mathrm{T}}$ is the rough approximation of the cross section for $\gamma \gamma$ absorption (${\sigma }_{\mathrm{T}}$ is the Thompson cross section) [51], n is the number density of the target photons that interact with high-energy photons and R is the emission radius. The X-ray detection displayed a very soft thermal spectrum with a peak energy around 100 eV, so the number density of soft photons n can be displaced by $U_X/E_{t,100\mathrm{eV}}$. The radiation energy density is

$$U_X=\frac{L_X}{4\pi R^{2}c}$$

(3)

One can derive the constraint of the emission radius R of neutrino production from the condition ${\tau }_{\gamma \gamma }>1$.

$$R<{\sigma }_{\gamma \gamma }{\displaystyle \frac{L_X}{4\pi c{E}_t}}\approx 7\times {10}^{15}{L}_{X,43}{E}_{t,100\mathrm{eV}}^{-1}\mathrm{cm}$$

(4)

Considering that the observed bolometric luminosities in the X-ray band are about ${10}^{42}$–${10}^{44}$ erg s${}^{-1}$ [30,31,32], we adopted $L_{X,43}\approx {10}^{43}$ erg s${}^{-1}$ for a modest estimation. This constraint is well compatible with a high-resolution radio observation [52] and some hidden source models, such as choked TDE jets [26] and AGN cores [30,37], or models where the neutrino production region is small (e.g., within the photosphere of TDEs).

To conclude, we searched for the high-energy gamma-ray emission from a special class of TDE candidates with strong accretion flares using Fermi-LAT survey data, especially the three TDEs likely associated with IceCube neutrinos, including AT2019dsg, AT2019fdr and AT2019aalc. No significant GeV emission was found from these TDEs during the neutrino arrival time, as well as other periods when Fermi-LAT normally operated for the three sources. Then, we performed a stacking analysis of all undetected sources in order to investigate a cumulative signal. Stacking the Fermi-LAT data of the below-threshold sources did not result in a detection either. The nondetection of high-energy emission in these TDEs implies that high-energy gamma-ray emission is seriously suppressed, possibly due to the $\gamma \gamma$ attenuation by soft photons in the source. We placed the emission radius of the neutrinos to be: $R<{10}^{16}{L}_{X,43}{E}_{t,100\mathrm{eV}}^{-1}$ cm.